Method and system of seismic data processing

ABSTRACT

The present invention relates to processing of seismic data. More specifically, the present invention relates to processing of low-cut filtered seismic data to reduce or suppress transient effects.

TECHNICAL FIELD

The present invention relates to processing of seismic data. Morespecifically, the present invention relates to low-cut filtering ofseismic data to reduce or suppress transient effects.

BACKGROUND OF THE INVENTION

In the field of seismic data processing, such as marine seismic dataprocessing, acquisition techniques include using low-cut filters toreduce low-frequency noise on the acquired data. Low-cut filtering mayhowever inadvertently or inevitably cause undesirable artefacts on theacquired data. In particular, the finite amplitude of a signal at thestart and at the end of the recoding time window may manifest in thefiltered data as unwanted distortions or “transient effects”. Any suchtransient effects may mask useful or important signatures of theacquired signal.

SUMMARY

In a first aspect of the present disclosure, there is provided acomputer-implemented method of low-cut filtering a seismic tracerecorded over a recording time window, the method comprising:

applying a causal low-cut filter to the seismic trace to generate firstfiltered data;

applying an anti-causal low-cut filter to the seismic trace to generatesecond filtered data;

truncating the first filtered data to generate first truncated databased on the recording time window;

truncating the second filtered data to generate second truncated databased on the recording time window;

removing the phase of the first truncated data;

removing the phase of the second truncated data;

selecting a portion of the first phase-removed truncated data togenerate first modified data, the selected portion of the firstphase-removed truncated data being associated with a later time intervalof the seismic trace;

selecting a portion of the second phase-removed truncated data togenerate second modified data, the selected portion of the secondphase-removed truncated data being associated with an earlier timeinterval of the seismic trace; and

generating a low-cut filtered seismic trace by combining at least thefirst modified data and the second modified data.

The step of removing the phase of the first truncated data may includeapplying a phase removal filter to the first truncated data. The phaseremoval filter may include an all-pass filter with a conjugate phase ofthe causal filter. Alternatively, the phase removal filter may include atime-reversed version of the causal filter.

The step of removing the phase of the second truncated data may includeapplying a phase removal filter to the second truncated data. The phaseremoval filter may include an all-pass filter with a conjugate phase ofthe anti-causal filter. Alternatively the phase removal filter mayinclude a time-reversed version of the anti-causal filter.

Applying a causal low-cut filter to the seismic trace to generate firstfiltered data may include computing discrete convolution of the seismictrace and the impulse response of the causal low-cut filter, and whereintruncating first filtered data may include removing convolutionartefacts arising from the discrete convolution. Truncating to removeconvolution artefacts may include truncating to the recording timewindow of the seismic trace.

Applying an anti-causal low-cut filter to the seismic trace to generatesecond filtered data may include computing discrete convolution of theseismic trace and the impulse response of the anti-causal low-cutfilter, and wherein truncating second filtered data may include removingconvolution artefacts arising from the discrete convolution. Truncatingto remove convolution artefacts may include truncating to the recordingtime window of the seismic trace.

The earlier interval of the seismic trace and the later interval of theseismic trace may each be a temporal half of the seismic trace.

The seismic trace may include an intermediate interval between theearlier interval and the later interval of the seismic trace, andwherein generating a low-cut filtered seismic trace may includecombining the first modified data, the second modified data and dataassociated with the intermediate interval.

The causal filter may be a minimum phase filter.

The anti-causal filter may be a maximum phase filter.

In one example, the anti-causal or causal low-cut filter includes a cutoff frequency of 2 Hz or less.

In one example, the anti-causal or causal low-cut filter includes anamplitude roll off of 12 dB per octave.

The anti-causal or causal low-cut filter may include any one of aone-dimensional filter, a two-dimensional F-K filter and anN-dimensional filter.

For a one dimensional filter, the independent variable of theone-dimensional filter may be space, frequency or voltage.

In a second aspect of the present disclosure, there is provided a systemconfigured for low-cut filtering a seismic trace, the system comprising:

an input for receiving the seismic trace;

one or more processing units configured to execute the method of thefirst aspect; and

an output for providing the low-cut filtered seismic trace.

In a third aspect of the present disclosure, there is provide anon-transitory machine-readable medium comprising instructions codedthereon for one or more processing units to execute the method of thefirst aspect.

Further aspects of the present disclosure and further embodiments of theaspects described in the preceding paragraphs will become apparent fromthe following description, given by way of example and with reference tothe accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

Embodiments and features of the present disclosure will be describedwith reference to the accompanying figures in which:

FIG. 1 illustrates an input sinusoidal time series (in solid lines) andthe times series filtered by a 2-Hz low cut filter;

FIG. 2 illustrates an example of a method of low-cut filtering accordingto the present disclosure;

FIG. 3 illustrates suppression or reduction of transient effects by acausal filter towards the end of the recording window;

FIG. 4 illustrates the filtered output of FIG. 3 having been partiallynulled and then having a time-reversed version of the causal filterapplied;

FIGS. 5A and 5B illustrate the impulse response and the frequencyresponse (amplitude and phase), respectively, of an example of a causalfilter;

FIG. 6 illustrates an example of towed streamer data;

FIG. 7A illustrates causally filtered and partially nulled streamerdata;

FIG. 7B illustrates anti-causally filtered and partially nulled streamerdata;

FIG. 7C illustrates the combined data illustrated in FIGS. 7B and 7C;

FIG. 8 illustrates the towed streamer data of FIG. 6 having been low-cutfiltered by a conventional method; and

FIG. 9 illustrates an example of a computer system suited forimplementing the computer-implemented method according to the presentdisclosure.

DETAILED DESCRIPTION OF THE INVENTION

Described herein is a method of processing of seismic data. Morespecifically, the present invention aims to low-cut filter seismic datawith reduced or suppressed transient effects. Also described herein arethe corresponding system and computer-readable medium which implementsuch a method.

The present disclosure is suited for use with processing marine seismicdata. However, the present disclosure may also have broader applicationin other fields in which finite length segments of time series arefiltered. It is also equally applicable to digitized functions of anyother independent variable besides time, for example, space orfrequency.

Transient Effects

In marine seismic surveys, broadband acquisition techniques sometimesuse very low low-cut field filters. In some instances, the instrumentfilters are “out”, or otherwise incorrectly configured, and the onlylow-cut filter effect may come from the hydrophone RC filter effect,which can be inadequate. As a result, significant low-frequency noise isoften recorded and requires some form of attenuation or manipulationduring data processing. For example, surface waves on the sea have adecaying pressure field at depth which is recorded by the hydrophones asa very low frequency (˜0.1 to 2 Hz) slowly moving (2 to 6 m/s) wavefield. The exact details of surface waves depend very much on theiramplitude, wavelength and direction of propagation relative to thestreamer.

Data acquisition captures data in a finite recording time window. Thefinite amplitude or value of a signal at the start and the end of therecording time window may manifest in the filtered data as unwantedtransient effects. To illustrate the transient effects due to low-cutfiltering data which has low-frequency components, consider a finitesegment of single-frequency data. FIG. 1 illustrates an input 102 whichis a 0.15 Hz sinusoidal time series. The horizontal axis represents timein seconds and the vertical axis represents data amplitude in anarbitrary scale. The time series lasts for 1.5 cycles and has non-zeroamplitudes at the start and the end of the time window. Also illustratedin FIG. 1 is the input 102 having been filtered as output 104 using a2-Hz low cut filter. The output 104 has zero (or almost zero) amplitudesexcept for the contribution from the ends of the time series 106, 108.The contribution contaminates the output for approximately 1 second ateach end within the duration of the time series. Because low-cut filtershave long responses, a simple low-cut filter for the removal of lowfrequency noise will contaminate a significant portion of the timeseries. Such distortions or transient effects at the ends of therecording time window may become significant if their amplitudes maskthe actual data.

The transient effects may be better understood mathematically. Intypical seismic measurements, the amplitudes at the beginning and theend of a seismic trace are expected to be zero or close to zero. This isbecause with a typical low-cut filter there is no energy anticipatedbefore the seismic source fires and negligible energy at the end of theseismic record. However, if the low filters are “out”, energy that isnot related to the seismic source, such as surface waves, is recordedand effectively has an abrupt start and end imposed by the recordingwindow. Given such a window function w(t) of time t with a window lengthof a,

$\begin{matrix}{{w(t)} = \{ \begin{matrix}1 & {{t} < \frac{a}{2}} \\0 & {{t} > \frac{a}{2}}\end{matrix} } & (0)\end{matrix}$

applied to pressure wavefield, p(t), we have an output:

u(t)=w(t)·p(t).  (0)

If we should then apply a digital filter, f(t), we will have theresulting data given by,

d(t)=f(t)*[w(t)·p(t)].  (0)

where * denotes convolution. To demonstrate the effect of the filter onthe windowed pressure function, consider taking the derivative withrespect to time,

$\begin{matrix}{{\frac{}{t}{d(t)}} = {{{f(t)}*{\frac{}{t}\lbrack {{w(t)} \cdot {p(t)}} \rbrack}} = {{f(t)}*{\lbrack {{\frac{w}{t} \cdot p} + {w \cdot \frac{p}{t}}} \rbrack.}}}} & (0)\end{matrix}$

Now if p(t) is smoothly varying then the second term in bracketsw·(dp/dt) will contribute only weakly. The first term in brackets(dw/dt)·p will be everywhere zero, except at the window ends so that,

$\begin{matrix}{{\frac{}{t}{d(t)}} \approx {{f(t)}*{\lbrack {{p(t)} \cdot ( {{\delta ( {t + {a/2}} )} - {\delta ( {t - {a/2}} )}} )} \rbrack.}}} & (0)\end{matrix}$

where δt) is the Dirac delta function. We invoke the sifting propertyand see that upon re-integration,

d(t)≈∫_(−∞) ^(+∞) f(t)dt*[p(−a/2)−p(+a/2)].  (0)

It may be concluded that, at the start and end of the recoding window,impulses proportional to p(−a/2) and −p(+a/2) respectively, will bepresent. Their shape is the time integral of the filter. The duration of∫_(−∞) ^(+∞)f(t)dt and the amplitudes of p(t) at the window ends areimportant because they determine how badly d(t) is contaminated.

From FIG. 1, it can be seen that low cut filtering is generallyeffective except for the transients generated at the trace ends withinthe recording window. The transient effects arise because filteringimplies that the trace is zero outside the recording window. One way toestimate the sinusoidal content in a window of limited length is toavoid using the zero sample values in the estimate.

Reducing Transient Effects

A method is disclosed to reduce or suppress the transient effects onlow-cut filtered seismic data of a finite recording time window. Thedisclosed method separately applies a causal filter and an anti-causalfilter to input data and combines selected portions of the causally andanti-causally filtered data to generate a full set of low-cut filtereddata.

In one general form, as schematically illustrated in FIG. 2, thedisclosed method 200 comprises essentially two streams of steps. Thefirst stream 200 a comprises: the step 202 of applying a causal low-cutfilter f(+t) to an input seismic trace d(t) 201 to generate firstfiltered data, the step 204 of truncating the first filtered datagenerate first truncated data based on the recording time window, thestep 206 of removing the phase of the first truncated data, the step 208of selecting a portion of the first phase-removed truncated data togenerate first modified data, the selected portion of the phase-removedtruncated data being associated with a later time interval of theseismic trace. The second stream 200 b comprises: the step 210 ofapplying an anti-causal low-cut filter f(−t) to the seismic trace d(t)to generate second filtered data, the step 212 of truncating the secondfiltered data to generate second truncated data based on the recordingtime window, the step 214 of removing the phase of the second truncateddata, the step 216 of selecting a portion of the second phase-removedtruncated data to generate second modified data, the selected portion ofthe second phase-removed truncated data being associated with an earliertime interval of the seismic trace, and the step 218 of generating alow-cut filtered output seismic trace by combining at least the firstmodified data and the second modified data.

A person skilled in the art would appreciate that the disclosed method200 does not require, to the extent possible, strict compliance to theorder in which the steps are introduced above or in FIG. 2. For example,in another general form of the method, the order of the first stream andthe second stream may be swapped or they may be executed in parallel.Furthermore, the skilled person would appreciate that the steps 208 and216 of selection may be alternatively implemented by nulling ordisregarding, or truncating a complementary portion of the relevantlyfiltered data. Similarly, the steps 204 and 212 of truncation may bereplaced by nulling a relevant portion of the filtered data.

The disclosed method may be computer-implemented, for example by acomputer system having an input for receiving the seismic trace, one ormore processing units configured to execute the disclosed method, and anoutput for providing the low-cut filtered seismic trace. The disclosedmethod may be coded as machine-readable instructions on a non-transitorymachine-readable medium for the one or more processing units to executethe method.

The reduction or suppression of the transient effects by the disclosedmethod may be illustrated through explanatory examples. FIG. 3illustrates the suppression or reduction of transient effects towardsthe end of the recording window. In FIG. 3, an output 304 (dashed line)is generated by applying a causal filter to a rectangle input function302 (solid line) having an arbitrary amplitude of 1 between arbitrarytime units 1 to 9 (solid line). While the causal filter causesdistortions (see upward spike 306) to the output 304 near the start ofthe rectangle input function, the causal filter causes no distortion orno significant distortion to the output up until the end of therectangle input function.

The application of a causal filter in the temporal domain on the seismictrace (that is, via convolution of the impulse response of causal filterand the seismic data) performs in a well-behaved manner up to the lastsample on the seismic trace because all the contributions are taken fromexisting samples. However, at the beginning of the trace the scenario isdifferent. There, contributions to output samples are taken implicitlyfrom samples prior to the first sample of the seismic trace so that thetime window at the start of the output trace that is equivalent inlength to the filter is distorted due to lack of valid contributions.For similar reasons, data beyond the end of the rectangle input function302 are also distorted (see downward spike 308). Still referring to FIG.3, the output 304 generated by applying a causal filter to the rectangleinput function 302 however provides little or no distortion up until theend of the rectangle input function.

Therefore, when a seismic trace is filtered by a causal filter, theportion of the filtered seismic data associated with the later intervalof the seismic trace can be undistorted or substantially undistorted.This undistorted or substantially undistorted portion of the filteredseismic data (e.g. the later half) may be selected. The selected portionof the causally filtered may be stored. The rest of the data portionsmay be truncated, nulled or otherwise disregarded. In other words, toreduce or suppress transient effects at the later interval of theseismic trace, corresponding input data may be first filtered by alow-cut causal filter. Thereafter, the portion of the filtered dataassociated with an earlier interval (e.g. the earlier half) of theseismic trace as well as the portion of the filter data beyond the endof the rectangle input may be truncated, nulled or otherwisedisregarded. The resulting modified data represent an undistortedportion (e.g. later half) of the low-cut filtered seismic trace.

In a similar manner, although not illustrated, if an anti-causal filteris applied to the rectangle function, the anti-causal filter causesdistortions to the output near the end of the rectangle input function,but no distortion or significant distortion to the output near the startof the rectangle input function. Therefore, when the seismic trace isfiltered by an anti-causal filter, the portion of the filtered seismicdata associated with the earlier interval of the seismic trace can beundistorted or substantially undistorted. This undistorted orsubstantially undistorted portion of the filtered seismic data (e.g. theearlier half) may be selected. The selected portion of the causallyfiltered data may be stored. The rest of the data portions may betruncated, nulled or otherwise disregarded. In other words, to reduce orsuppress transient effects at the earlier interval of the seismic trace,corresponding input data may be first filtered by a low-cut anti-causalfilter. Thereafter, the portion of the filtered data associated with alater interval (e.g. the later half) of the seismic trace as well as theportion of the filter data beyond the start of the rectangle input maybe truncated, nulled or otherwise disregarded. The resulting modifieddata represent an undistorted portion (e.g. earlier half) of the low-cutfiltered seismic trace.

To generate a full low-cut filtered seismic trace with reduced orsuppressed transient effects, the modified data representing the earlierinterval (e.g. the earlier half) and the later interval (e.g. the laterhalf) may be combined, for example by concatenation (if portions offiltered data are truncated) or addition (if portions of filtered dataare nulled).

Applying a filter, such as in steps 202 and 210, involves digital signalprocessing. An example of applying a filter is to take the discreteconvolution of an underlying set of data (e.g. a seismic trace) with theimpulse response of the filter. The discrete convolution of a seismictrace array of length M (over a recording time window, or between thestart of the window, “tmin”, and the end of the window, “tmax”) and afilter array of length N generally results in an output array of lengthN+M−1. Artefacts associated with boundary effects may be introduced justoutside the boundaries of the recording time window (i.e. at t<tmin andt>tmax). It is therefore necessary to truncate the output array to, forexample, the recording time window of length M, or null the output arrayoutside the recording time window of length M, or otherwise remove theboundary-associated artefacts. If these boundary-associated artefactsare not removed, any further discrete convolution (e.g. during phaseremoval by a phase removal filter array as further described below)would re-introduce unwanted energy within the recording time window.Truncation steps 204 and 212 are intended to remove theboundary-associated artefacts.

In some arrangements, the modified data representing the earlierinterval may be the earliest one-third of the seismic trace (whereas thelater two-thirds are disregarded), and the modified data representingthe later interval may be the latest one-third of the seismic trace(whereas the earlier two-thirds are disregarded). In these arrangements,data representing an intermediate interval (e.g. middle one-third) ofthe filtered seismic trace (e.g. filtered by any low-cut filter, whetheror not causal or anti-causal) may be combined with the modified data(e.g. the earliest and latest one-thirds) to generate a full low-cutfiltered seismic trace with reduced or suppressed transient effects.Whether two portions or three portions are combined to generate a fulllow-cut filtered seismic trace, their corresponding intervals may be ofdifferent and arbitrary lengths subject to two constraints: a) Thelength of one such portion to be filtered is at least twice the filterlength, and, b) the point of division between the earlier and laterparts of one portion must be at least one filter length from both endsof the portion to be filtered.

In general, the phase spectrum of the causal and/or anticausal filterimposes an unwanted or undesirable phase on the filtered data. In method200, steps 206 and 214 are to remove the phase of the truncated data. Toremove the phase effects of the filter on the truncated data, steps 206and 214 may each apply a phase removal filter. The phase removal filtermay include an all-pass filter with a conjugate phase of the causalfilter, or a time-reversed version of the causal filter. FIG. 4illustrates the filtered output 304 of FIG. 3 having been (1) nulled and(2) applied with a time reversed version of the causal filter for phaseremoval. The phase-removed result is illustrated as line 402.

FIGS. 5A and 5B show the impulse response 502 and the frequency response(amplitude 504 and phase 506) of an example of a causal filter suitedfor use in the present disclosure. The illustrated example is a RClow-cut filter with a 2 Hz cut off frequency at −6 dB and a 12 dB peroctave roll off. An anti-causal filter may be effectively constructedusing a causal filter, such as that represented in FIGS. 5A and 5B. Inparticular, the anti-causal filter may be implemented by (1)time-reversing the input data, (2) applying a causal filter to thetime-reversed input data and (3) time-reversing the filteredtime-reversed input. The causal filter may be a minimum phase filter andthe anti-causal filter may be a maximum phase filter. The causal andanti-causal filters may be a one-dimensional filter (in which theindependent variable of the one-dimensional filter is space, frequencyor voltage), a two-dimensional F-K filter or an N-dimensional filter.

Accordingly, the disclosed method enables low-cut filtering a truncatedsegment of data without creating transient effects. For all but theearliest interval of the time window, this is achieved by convolution,truncation and correlation. For all but the latest interval of data,this is achieved by correlation, truncation and convolution. The tworesulting sets of data may then be combined (in some cases with datarepresenting an intermediate interval of the window) to providetransient-suppressed low-cut filtered data.

Application Example

FIG. 6 illustrates towed streamer data 600 obtained in a marine surveyacquisition. In some implementations, compressed air is released togenerate a seismic source. The response is measured along a towedstreamer containing hydrophones. In some implementations, particlevelocity is measured by geophones or acceleration is measured byaccelerometers in addition to the pressure measurement by hydrophones.The streamer data 600 show received signal amplitude over space(measured in offset distance from the seismic source) and time. Thevertical axis represents time (spanning a 5-second interval representingthe recording time window) increasing in the downward direction and thehorizontal axis represents offset (spanning a 6-km distance) increasingin the right direction. FIG. 6 shows fine lines, for example within theregion identified by ellipse 602, oriented diagonally substantially fromthe top left to the bottom right of the plot arising from, for example,reflections off the seabed and reflections from below the seabed. FIG. 6also shows substantially vertical lines, for example within the regionidentified by ellipse 604 (that is, along the direction of the timeaxis) across the entire plot of FIG. 6. The substantially vertical linesrepresent unwanted low frequency noise due to, for example, surfacewaves at or near each offset distance. The small deviation of theselines from vertical represents the movement of the surface waves.

To low-cut filter the streamer data 600 such that signals manifesting asvertical lines identified by ellipse 604 can be reduced or suppressed,the disclosed method 200 may for example be applied. Specifically, instep 202, the streamer data 600 is filtered via discrete convolution bya low-cut causal filter, such as one having responses shown in FIGS. 5Aand 5B. In step 204, discrete convolution artefacts are removed bytruncation of the causally filtered data to the recording time window(i.e. between tmin and tmax) before application of a phase removalfilter. In step 206, the phase of the causally filtered data is removedby applying a phase removal filter. In step 210, the streamer data 600is filtered via discrete convolution by a low-cut anti-causal filter,such as one based on the low-cut causal filter of FIGS. 5A and 5B, butwith each of the input and the output time-reversed. In step 212,discrete convolution artefacts are removed by truncation of theanti-causally filtered data to the recording time window (i.e. betweentmin and tmax) before application of a phase removal filter. In step214, the phase of the anti-causally filtered data is removed by applyinga phase removal filter.

In step 208, a portion of the causally filtered data, representing anearlier half interval of the streamer data, is truncated, nulled orotherwise disregarded. That is, a later half interval of the causallyfiltered data is selected as first modified data. The later halfinterval 702 of the causally filtered data is illustrated in FIG. 7A.The low-frequency noise (see lines 604) as well as transient distortionsin this portion of the causally filtered data have been suppressed.Similarly, in step 216, a portion of the anti-causally filtered data,representing a later half interval of the streamer data 602, istruncated, nulled or otherwise disregarded. That is, an earlier halfinterval of the anti-causally filtered data is selected as secondmodified data. The earlier half interval 704 of the anti-causallyfiltered data is illustrated in FIG. 7B. Again, the low-frequency noise(see lines 604) as well as transient distortions have been suppressed.In step 210, the earlier half interval 704 of the anti-causally filtereddata and the later half interval 702 of the causally-filtered data arecombined by concatenation to generate low-cut filtered streamer data706. The generated low-cut filtered streamer data 706 is shown in FIG.7C.

For comparison purposes, FIG. 8 illustrates the result of applying aconventional low-cut filter to the streamer data 600 to remove thelow-frequency noise (see lines 604) without application of method 200.While the low-frequency noise is removed, significant distortions arepresent at the start 708 (towards the top of the plot of FIG. 8) and theend 710 (towards the bottom of the plot of FIG. 9) of the recording timewindow. Such distortions may mask useful information. In contrast, thestreamer data which have been low-cut filtered according to the presentdisclosure and illustrated in FIG. 7C do not present these significantdistortions.

One implementation of the disclosed method performed on streamer data600 may be summarized in the following pseudo-code:

! --- Obtain necessary data f(t) = Compute causal low cut filter d(t) =read(data trace) tmin = start_time(d(t)) tmax = end_time(d(t)) tmid =(tmax+tmin)/2 ! --- compute the front end and tail end results !  ---front end portiona(t)=truncate(f(+t)*truncate(f(−t)*d(t),tmin,tmax),tmin,tmid) !  ---tail end portionb(t)=truncate(f(−t)*truncate(f(+t)*d(t),tmin,tmax),tmid,tmax) ! --- putthe two halves together in g and output g(t) = a(t)+b(t) save(g(t))

In the above pseudo code, d(t) represents streamer data 600, a(t)represents the earlier half interval 704 of the anti-causally-filtereddata, b(t) represents the later half interval 702 of thecausally-filtered data, f(+t) and f(−t) are the time-forward andtime-reverse filter functions, respectively, representing a causalfilter and an anti-causal filter, respectively, and g(t) represents thelow-cut filtered streamer data 706. tmin, tmid and tmax represent thebeginning, the mid-point and the end of the recording time window. Theasterisk * denotes a convolution operation. The time reversal indicatedin f(−t)* is equivalent to a correlation operation (i.e. convolutionwith one time series reversed. That is, f(−t)* denotes correlation withf(+t). Using the 5-second recording time window of FIG. 6 as an example,tmin corresponds to t=0 second and tmax corresponds to t=5 second.

FIG. 2 corresponds to the pseudo-code above. For the earlier interval(i.e. “front end”) of the seismic trace, d(t) is (1) discretelyconvolved with f(−t), (2) truncated to the time-recoding window, (3)discretely convolved with f(+t), and (4) truncated to the earlier halfinterval of the recording time window to generate a(t). For the laterinterval (i.e. “tail end”) of the seismic trace, d(t) is (1) discretelyconvolved with f(+t), (2) truncated to the time-recoding window, (3)discretely convolved with f(+t), and (4) truncated to the later halfinterval of the recording time window to generate b(t). The low-cutfiltered streamer data g(t) is then generated by combining a(t) and b(t)by, for example, concatenation to form filtered data spanning the fullinterval of the recording time window.

In one example, the pseudo code may be implemented as a series of matrixoperations. Specifically, the filter operation may be represented inconvolutional form:

$\begin{matrix}{{F = \begin{bmatrix}f_{0} & \ldots & \ldots & \ldots & \ldots & \ldots \\f_{1} & f_{0} & \ldots & \ldots & \ldots & \ldots \\\ldots & f_{1} & f_{0} & \ldots & \ldots & \ldots \\\ldots & \ldots & f_{1} & f_{0} & \ldots & \ldots \\\ldots & \ldots & \ldots & f_{1} & f_{0} & \ldots \\\ldots & \ldots & \ldots & \ldots & f_{1} & f_{0}\end{bmatrix}},} & (1.1)\end{matrix}$

and the truncation operations in matrix form,

$\begin{matrix}{{{T_{1} = \begin{bmatrix}\ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\\ldots & 1 & \ldots & \ldots & \ldots & \ldots \\\ldots & \ldots & 1 & \ldots & \ldots & \ldots \\\ldots & \ldots & \ldots & 1 & \ldots & \ldots \\\ldots & \ldots & \ldots & \ldots & 1 & \ldots \\\ldots & \ldots & \ldots & \ldots & \ldots & \ldots\end{bmatrix}},{T_{a} = \begin{bmatrix}\ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\\ldots & 1 & \ldots & \ldots & \ldots & \ldots \\\ldots & \ldots & 1 & \ldots & \ldots & \ldots \\\ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\\ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\\ldots & \ldots & \ldots & \ldots & \ldots & \ldots\end{bmatrix}},{and}}{{T_{b} = \begin{bmatrix}\ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\\ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\\ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\\ldots & \ldots & \ldots & 1 & \ldots & \ldots \\\ldots & \ldots & \ldots & \ldots & 1 & \ldots \\\ldots & \ldots & \ldots & \ldots & \ldots & \ldots\end{bmatrix}},}} & (1.2)\end{matrix}$

so that the disclosed method 200 may be described as,

(T _(a) FT ₁ F ^(T) +T _(b) F ^(T) T ₁ F)d=g,  (1.3)

where F^(T) represents the transpose of F.

As will be appreciated by a skilled person, there are a wide range oforders in which the operations in Equation (1.3) may be implemented. Forinstance, addition is associative and commutative. Multiplication isassociative and left/right distributive over addition. Accordingly, thefollowing forms of evaluation of g, for example, are each equivalent toEquation (1.3):

g=(T _(a) FT ₁ F ^(T) +T _(b) F ^(T) T ₁ F)d

g=(T _(a) FT ₁ F ^(T))d+(T _(b) F ^(T) T ₁ F)d.

g=[T _(a)(FT ₁ F ^(T))+T _(b)(F ^(T) T ₁ F)]d

It should be apparent that the method 200, if implemented by matrixoperations, has been explicitly described as:

T _(a)(F(T ₁(F ^(T) d)))+T _(b)(F ^(T)(T ₁(Fd)))=T _(a) FT ₁ F ^(T) d+T_(b) F ^(T) T ₁ Fd=g.  (1.4)

It should also be apparent to a skilled person that although thebrackets in Equation (1.4) explicitly order the evaluation sequence,they are not strictly necessary since matrix algebra is by conventionevaluated left to right. Further. it should also be apparent that theoperator F is equivalent to a convolution operation in the time domainor a multiplication operation in the Fourier or frequency domain.

In some cases meta-languages may be used to implement the disclosedmethod. For example in ProMAX/SeiSpace, the disclosed method may beexecuted by the following sequence:

Javaseis Data Input <- d User-Defined Filter  (correlation) User-DefinedFilter  (convolution) JavaSeis Data Output -> a ----- Add Flow Comment----- Javaseis Data Input <- d User-Defined Filter  (convolution)User-Defined Filter  (correlation) JavaSeis Data Output -> b ----- AddFlow Comment ----- Javaseis Data Input <- a Javaseis Data Combine <- bTrace Math (sum trace pairs) JavaSeis Data Output -> g

Computer Implementation

As noted, the disclosed method may be implemented using a computersystem. In general, as depicted in FIG. 9, the corresponding systemincludes one or more of computer processing systems 920 a-920 d.Computer processing systems 920 a-920 d may be communicatively coupledvia a communications network 901. Communications network 901 mayinclude, for example, any one or more of a wired, wireless, satellite,microwave and fibre optic communications network. The computerprocessing systems may be, respectively, a computer server 920 a, adesktop computer 920 b, a laptop computer 920 c, or a computer serversystem 920 e. For example, method 200 may be executed on computer server920 a, whereas streamer data for processing may be stored in computerserver system 920 e. As another example, several computer processingsystems, such as computer processing systems 920 a and 920 d may executethe method cooperatively using distributed computing techniques.

Each of computer processing systems 920 a-920 d includes at least oneprocessing unit 922 which may be a single computational processingdevice (e.g. a microprocessor or other computational device) or aplurality of computational processing devices. Through a communicationsbus 924, processing unit 922 is in data communication with a systemmemory 926 (e.g. a read only memory storing a BIOS for basic systemoperations), a volatile memory 928 (e.g. random access memory such asone or more DRAM modules), and a non-transient memory 930 (e.g. one ormore hard disk drives, solid state drives, flash memory devices andsuchlike). Instructions and data to control operation of processing unit922 are stored on system, volatile, and/or non-transitory memories 926,928, and 930. For example, computer codes for executing method 200 maybe stored in or downloaded into memories 926, 928 and 930 and/orgenerated by processing unit 922 based on instructions stored in ordownloaded into memories 926, 928 and 930.

At least one computer processing systems 920 a-d may also include one ormore input/output interfaces 932 which allow system 920 to interfacewith a plurality of input/output devices 934 and 936, or via one or moreports 938. As will be appreciated, a wide variety of input/outputdevices may be used depending on the device/system/apparatus inquestion, for example keyboards, pointing devices, touch-screens,touch-screen displays, displays, microphones, speakers, hard drives,solid state drives, flash memory devices and the like. Computerprocessing system 920 also includes one or more network communicationsinterfaces 940, such as Network Interface Cards, modems and the like,allowing for wired and/or wireless connection to communications network901. For example, communications network 901 may include am intranet.Alternatively, communications network 901 may include the Internet whichenables communications between a server and a client remote from eachother.

Computer processing system 920 stores in memory and runs one or moreapplications allowing operators to locally or remotely operate or managesystem 920. Such applications will typically include at least anoperating system such as Microsoft Windows, Apple OSX, Unix, Linux,Apple iOS, Google Android, or other operating system.

Communication with communications network 901 (and other devices,apparatuses, servers, apparatuses connected thereto) may be by theprotocols set out in the layers of the Open Systems Interconnection(OSI) model of computer networking. For example, applications/softwareprograms being executed by computer processing system 920 maycommunicate using one or more transport protocols, e.g. the TransmissionControl Protocol (TCP) or the User Datagram Protocol (UDP). Alternativecommunications protocols may, of course, be used. For data transfertasks, systems 902 a-d may use protocols such as the File TransferProtocol (FTP).

While FIG. 9 provides a general overview of suitable computer processingsystems, it should be appreciated that the server and user devicesdescribed herein may be of alternative system types. Further, and asnoted, each different system may have different I/O interfaces and I/Odevices, different communications interfaces, and/or different softwareapplications installed. Further, the disclosed method may be partially,additionally or alternatively implemented using customized hardware,such as one or more Digital Signal Processors (DSPs), Field ProgrammableGate Arrays (FPGAs) or Application Specific Integrated Circuits (ASICs).

It will be understood that the invention disclosed and defined in thisspecification extends to all alternative combinations of two or more ofthe individual features mentioned or evident from the text or drawings.For example, step 202 may use a causal filter not characterized by theresponses shown in FIGS. 5A and 5B. As noted, steps 204 and 206, orsteps 202 and 204, may be swapped. All of these different combinationsconstitute various alternative aspects of the invention.

As used herein, except where the context requires otherwise, the term“comprise” and variations of the term, such as “comprising”, “comprises”and “comprised”, are not intended to exclude further additives,components, integers or steps.

As used herein, except where the context requires otherwise, terms suchas “first”, “second” and “third” are used arbitrarily to distinguishbetween like elements such terms describe, and do not necessarily denoteorder or timing, or the preferred order timing of such elements.

What is claimed is:
 1. A computer-implemented method of low-cutfiltering a seismic trace recorded over a recording time window, themethod comprising: applying a causal low-cut filter to the seismic traceto generate first filtered data; applying an anti-causal low-cut filterto the seismic trace to generate second filtered data; truncating thefirst filtered data to generate first truncated data based on therecording time window; truncating the second filtered data to generatesecond truncated data based on the recording time window; removing thephase of the first truncated data; removing the phase of the secondtruncated data; selecting a portion of the first phase-removed truncateddata to generate first modified data, the selected portion of the firstphase-removed truncated data being associated with a later time intervalof the seismic trace; selecting a portion of the second phase-removedtruncated data to generate second modified data, the selected portion ofthe second phase-removed truncated data being associated with an earliertime interval of the seismic trace; and generating a low-cut filteredseismic trace by combining at least the first modified data and thesecond modified data.
 2. The method of claim 1 wherein the step ofremoving the phase of the first truncated data includes applying a phaseremoval filter to the first truncated data.
 3. The method of claim 2wherein the phase removal filter includes an all-pass filter with aconjugate phase of the causal filter.
 4. The method of claim 2 whereinthe phase removal filter includes a time-reversed version of the causalfilter.
 5. The method of claim 1 wherein the step of removing the phaseof the second truncated data includes applying a phase removal filter tothe second truncated data.
 6. The method of claim 5 wherein the phaseremoval filter includes an all-pass filter with a conjugate phase of theanti-causal filter.
 7. The method of claim 5 wherein the phase removalfilter includes a time-reversed version of the anti-causal filter. 8.The method of claim 1 wherein applying a causal low-cut filter to theseismic trace to generate first filtered data includes computingdiscrete convolution of the seismic trace and the impulse response ofthe causal low-cut filter, and wherein truncating first filtered dataincludes removing convolution artefacts arising from the discreteconvolution.
 9. The method of claim 8 wherein truncating to removeconvolution artefacts includes truncating to the recording time windowof the seismic trace.
 10. The method of claim 1 wherein applying ananti-causal low-cut filter to the seismic trace to generate secondfiltered data includes computing discrete convolution of the seismictrace and the impulse response of the anti-causal low-cut filter, andwherein truncating second filtered data includes removing convolutionartefacts arising from the discrete convolution.
 11. The method of claim10 wherein truncating to remove convolution artefacts includestruncating to the recording time window of the seismic trace.
 12. Themethod of claim 1 wherein the earlier interval of the seismic trace andthe later interval of the seismic trace are each a temporal half of theseismic trace.
 13. The method of claim 1 wherein the seismic traceincludes an intermediate interval between the earlier interval and thelater interval of the seismic trace, and wherein generating a low-cutfiltered seismic trace includes combining the first modified data, thesecond modified data and data associated with the intermediate interval.14. The method of claim 1 wherein the causal filter is a minimum phasefilter.
 15. The method of claim 1 wherein the anti-causal filter is amaximum phase filter.
 16. The method of claim 1 wherein the anti-causalor causal low-cut filter includes a cut off frequency of 2 Hz or less.17. The method of claim 1 wherein the anti-causal or causal low-cutfilter includes an amplitude roll off of 12 dB per octave.
 18. Themethod of claim 1 wherein the anti-causal or causal low-cut filterincludes a one-dimensional filter.
 19. The method of claim 18 whereinthe independent variable of the one-dimensional filter is space,frequency or voltage.
 20. The method of claim 1 wherein the anti-causalor causal low-cut filter includes a two-dimensional F-K filter.
 21. Themethod of claim 1 wherein the anti-causal or causal low-cut filterincludes an N-dimensional filter.
 22. A system configured for low-cutfiltering a seismic trace, the system comprising: an input for receivingthe seismic trace; one or more processing units configured to executethe method of any one of claims 1-21; an output for providing thelow-cut filtered seismic trace.
 23. A non-transitory machine-readablemedium comprising instructions coded thereon for one or more processingunits to execute the method of any one of claims 1-21.